Height
Claim
The shape stands r(2 + 3√2)/2 tall, about 3.1213r.
Construction
Put the head circle's centre at the origin and stand the shape upright. The diamond's centre sits directly below at distance r√2, which is the lattice pitch and is proved separately.
The highest point is the top of the head, r above the centre. The lowest points are the two bottom corners of the diamond — not the bottom of its lower arc, which is concave and curves back up between them.
Proof
The diamond's corners lie at distance r from its centre, at 45° to the axis, so the two lower corners are r/√2 = r√2/2 below it. The diamond's centre is r√2 below the head's centre, putting the lowest point 3r√2/2 below the head's centre.
h = r + 3r√2/2 = r(2 + 3√2)/2 ≈ 3.12132r
The common first answer is r(2 + √2) ≈ 3.414r, which takes the bottom of the lower arc as the lowest point. The arc's lowest reach is r(2√2 − 1) ≈ 1.828r below the head's centre, well above the corners at 2.121r.
Note
This is the constant that shows up in every physical build. Where did it first bite?
Open questions
- Whether the library should carry the exact form or the decimal in running text. The pages currently mix them.
Checks
| What | How | Source |
|---|---|---|
| Height of the exact path | Sampled bounding box converges on the predicted 312.132: 312.1208 at 20,000 points, 312.1308 at 200,000; the residual is sample spacing at the extreme | _src/core.py |
| Where the lowest point is | Extreme y occurs at x = ±70.70, the corners; the arc bottom reaches only 182.84 | _src/core.py |
| Aspect | h/w = 1.5606 measured, (2 + 3√2)/4 = 1.56066 predicted | reference/geometry-constants.md |